Question

find out the burnt out speed for rocket system with variable mass .

Answer 1

Applying the conservation of momentum for an isolated system:

pt=mvpt=mv

pt+Δt=(m−dm)(v+dv)+dm(v−ue)pt+Δt=(m−dm)(v+dv)+dm(v−ue)

dpdt=mv+mdv−vdm+vdm−uedm−mvdt=0dpdt=mv+mdv−vdm+vdm−uedm−mvdt=0

0=mdv−uedmdt0=mdv−uedmdt

dv=−uedmmdv=−uedmm

that is the classical Tsiolkovsky rocket equation, that can be integrated ∫tt0∫t0t:

Δv=uelnm0mΔv=ueln⁡m0m

Where ueue is the velocity of the gases that are coming out from the nozzle.

In other books the same equation is obtained from Newton's Law:

mdvdt=∑iFextimdvdt=∑iFiext

where the external forces are basically just the thrust (simplest case) that is given by:

T=m˙ue+Ae(pe−pa)=m˙cT=m˙ue+Ae(pe−pa)=m˙c

where AeAe is the area of outflow for the nozzle, pepethe outflow pressure, papa the ambient pressure (hence c=ue+Ae(pe−pa)m˙c=ue+Ae(pe−pa)m˙ is the equivalent velocity)

substituting:

mdvdt=m˙cmdvdt=m˙c

being m˙=−dmdtm˙=−dmdt because of the mass loss we get:

dv=−cdmmdv=−cdmm

that is basically the same equation but with the equivalent velocity instead of the real convective gas velocity